For every time a geometry student asks, “When am I ever going to use this in real life?” Well, if your child ever asks you to make her a Harley Quinn costume, and there is no pattern, so you have to draft your own, you will need to find the sides of a square using the measurement of the diagonal…

[I]f you need to have a square patchwork of different colored fabrics which line up on diagonal points for a specific measurement so that you have four colored diagonal squares from the shoulder to just below the waist, you would need to find the measurement of the four equal sides of each square. Then you would add seam allowances so you could cut the squares out of the different colored fabrics and sew them together in exact lines to line up just right so you could make a top that looks like the top the character wears. And since this character is only a cartoon character who has been made into a little doll, not many people out there in the world have yet attempted an actual costume to be worn by a real live girl. Of course, a person could just take a pencil and a ruler and draw squares, but without using math, that person could not put together a patchwork of colored fabric squares with this result.

This mathematical trick was not part of my Pi Day magic show but probably should have been. I first read about this trick in one of Martin Gardner‘s books when I was a teenager, and it’s amazing how impressive this appears when performed. I particularly enjoy stumping my students with this trick, inviting them to figure out how on earth I pull it off.

Here’s a video of the trick, courtesy of Numberphile:

Summarizing, there’s a way of quickly determining given the value of if is a positive integer less than 100:

The ones digit of will be the ones digit of .

The tens digit of can be obtained by listening to how big is. This requires a bit of memorization (and I agree with the above video that the hardest ones to quickly determine in a magic show are the ones less than and the ones that are slightly larger than a billion):

Here’s a standard joke involving representing numbers in different bases.

Q: If only DEAD people understand hexadecimal, then how many people understand hexadecimal?

A: 57,005.

The joke, of course, is that can be considered a number written in base 16, using the usual convention , , , , , and . In other words, can be converted to decimal as follows:

.

After I heard this joke, I wondered just how many English words can be formed using only the letters A, B, C, D, E, and F so that I could make a subtle joke on a test. To increase the length of my list, I also allowed words that included the letters O (close enough to a 0), I (close enough to 1), and/or S (close enough to 5). However, I eliminated words that start with O (since a numeral normally doesn’t start with 0) and/or end in S (the plural version of these words are easily formed).

So I wrote a small program to search the dictionary that I have on my computer. The unabridged list follows, with words beginning with a capital letter (such as names or places) listed at the bottom. I emphasize that this list is unabridged, as there are several words on this list that I wouldn’t place on a test for obvious reasons: I would never ask my class to convert the base-10 numeral 721,077 into hexadecimal just so they can obtain the answer of .